Keep track of how many times a recursive function was called

Answer 1

Another approach, since you produce all the numbers, is to use a generator.

The last element is your number n reduced to a single digit number and to count how many times you have iterated, just read the length of the array.

const digits = [...to_single_digit(39)];
console.log(digits);
//=> [27, 14, 4]

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Final thoughts

You may want to consider having a return-early condition in your function. Any numbers with a zero in it will return zero.

singleDigit(1024);       //=> 0
singleDigit(9876543210); //=> 0

// possible solution: String(n).includes('0')

The same can be said for any numbers made of 1 only.

singleDigit(11);    //=> 1
singleDigit(111);   //=> 1
singleDigit(11111); //=> 1

// possible solution: [...String(n)].every(n => n === '1')

Finally, you didn't clarify whether you accept only positive integers. If you accept negative integers then casting them to strings can be risky:

[...String(39)].reduce((x, y) => x * y)
//=> 27

[...String(-39)].reduce((x, y) => x * y)
//=> NaN

Possible solution:

const mult = n =>
  [...String(Math.abs(n))].reduce((x, y) => x * y, n < 0 ? -1 : 1)

mult(39)
//=> 27

mult(-39)
//=> -27